8.3 Divide & Conquer for tridiagonal A

Transcription

1 8 8.3 Dvde & Conqer for trdagonal A A dvde and conqer aroach for cotng egenvales of a syetrc trdagonal atrx. n n n a b b a b b a dea: Slt n two trdagonal atrces and. Cote egenvales of and. Recover the orgnal egenvales of as ertrbatons. Reeat recrsvely.

2 9 Slttng of v : vv : ~ Set A: Generate zeros at the sb/serdagonal entres n the ddle of * * ) :, : ( ~!, a b b a a b b a b vv Rank- ertrbaton of

3 Relaton between and,?,, Asse, that we know the egenvales and egenvectors of and. How can we get the egenars of? Note, that s a rank- ertrbaton of dag(, ). Recover the orgnal egenvales as ertrbatons of egenvales of and. vv vv vv vv ~ ~

4 Cotng the egenvector Hence, we need to cote the egenvales of a atrx of the for dagonal + rank- : D vv ~ ~ Let λ and be an egenar of D + ρvv. hen t holds ~~ D vv D v ~ v~ const D v~ Hence, f we know λ, then we drectly get the egenvector.

5 Egenvales as Zeros ~ ~ ~ ~ ) ( ~ ~ ~ ~ ~ ~ ~ n n d v d v v D v f v v D v v v v D D v Frtherore, we get the eqaton se Newton s ethod, to deterne the zeroes of fncton f(λ) hese zeroes are the egenvales of and therefore also of. Reeat recrsvely for and. vv D ~ ~

6 Zeros and oles of f(λ): λ λ λ 3 λ 4 d d d 3 d 4 d

7 8.4 Algorths for cotng a few egenars: Vector teraton: x ( k ) A x k () k () A x v egenvector to egenvale wth ax absolte vale Easy to arallelze (only Ax), bt slow convergence! Only λ ax! Sbsace teraton: Aly the sae dea to set of vectors () =(x (),,x () ) Consder egenvales of (k)h A (k) and then relace (k) by A (k) nverse teraton: Aly vector teraton on shfted roble (A σ ) - for cotng the egenvector nearest to σ. Exensve! ll-condtoned lnear syste! 4

8 5 Raylegh Qotent teraton ; ; ) ( ; y v y v A y y y v Raylegh Qotent teraton: Start wth vector y and real ρ and reeat: Fast convergence, bt ncertan to whch egenvale we wll converge. Exensve! ll-condtoned! nverse teraton wth relacng the shft σ by the newest egenvale estate. y: new egenvector estate leads to new egenvale estate: ) ( ) ( y v y y y y A y y y y A y y y Ay y new

9 8.5 Arnold (Lanczos) for sarse A se the transforaton on Hessenberg (trdagonal) for descrbed for GMRES. Cote the egenvales of the sall Hessenberg atrx and se the as aroxatons for the egenvales of the orgnal atrx. By Arnold Orthogonalzaton of the Krylov sbsace (b,ab,a b, ) we get the relaton A j j k h k, j k ~ j j k h k, j k A H, H, h, A ~ Egenvales of H, as aroxatons for A. (Sall h +, good aroxaton). Good aroxaton for extree egenvales 6

10 8.6 Jacob-Davdson for sarse A dea: - No Krylov sbsace - choose sbsace relatve to egenvale we are lookng for - nclde recondtonng; Startng ont: Consder egenvale aroxatons derved by V H AV for sbsace relatve to V. he egenars of V H AV are sed as aroxatons to soe egenvales of A How to choose new sbsace V + wth addtonal vector sch that the new aroxaton for secal egenvales s strongly roved? For frst egenar aroxaton and t =( H A )/( H ), we try to rove these aroxatons by sall correctons and t to get better estates + and t +t A(! ) ( t t)( ), 7

11 8 Jacob-Davdson t t A ), )( ( ) ( t t A t t A gnore correcton t of second order se orthogonal rojecton wth fro the left. hs leads to H H H t A t A t A t A t A H H H H H =

12 Jacob-Davdson For new aroxaton we have to solve A t A t P H A t P r or A r H ~ At gets ll-condtoned for t near egenvale, bt P s a rojecton orthogonal to the near snglar vector! A ~ s snglar, bt lnear syste s stll solvable. Relace ll-condtoned by snglar syste. 9

13 Jacob-Davdson V New egenvector estate + + also leads to new egenvale estate t + ( +H A + )/( +H + ). Choose the new estate + to enlarge the sbsace V by the new vector to V +. Cote egenars of V +H AV +. Reeat ths ste a few tes. Restart the whole rocess wth last best aroxaton as startng vector, res. -d sbsace V. Advantages: Allows to cote also nner egenvales wthot solvng ore and ore ll-condtoned robles lke Raylegh Q.

14 Jacob-Davdson V Man ste: Solve lnear syste P A t P r or P ( At ) r aroxately. ~ herefore, we se a few stes of recondtoned cg or GMRES. Precondtoner: M - recondtoner for A PM - P recondtoner for PAP n each teraton ste we have to ltly wth A, wth P, and solve n M. Sle recondtoner: M = dag(a) Better recondtoner: SPA or MSPA

15 8.7 Bsecton for cotng egenvales of a trdagonal atrx Observaton: he characterstc olynoal of a trdagonal atrx can be evalated va the atrx entres n for of a seqence of olynoals wth ncreasng degree: n n n n n det det ) ( 3 n 3,4,..., ), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( n

16 he seqence of olynoals s a Str chan:. All have only sngle zeros. sgn( n- (a)) = - sgn( n (a)) for all real zeros of n (x) 3. For =,,,n-: + (a) - (a) < for all real zeros of (x) 4. he olynoal (x) does not change st sgn Defne w(a):= # sgn changes n (a), =,,n. t holds: w(a) = # zeros of n (x) for x<a. Consder egenvales ordered λ < λ < < λ n- < λ n. We want to fnd λ. t holds: λ < a w(a) = [ # zeros left of a ] >= 3

17 Str Condton: P + - a At all zeros of the neghbors - and + st have dfferent sgn. 4

18 Bsecton Algorth: Choose an nterval =[a,b ] whch contans λ. herefore: w(b ) >= and w(a ) <. Evalate the olynoal seqence for a=(a +b )/ and cont the sgn changes n the seqence (a) w(a). f w(a) >= : Relace n b by a Otherwse: Relace n a by a. Generates convergenng seqence of saller and saller ntervals that garanteed contan the egenvale λ. Advantages: - can be easly arallelzed - can be sed wth hgh or low accracy 5

19 8.8 MR 3 for trdagonal atrces dea: se nverse teraton for cotng the egenvectors of a trdagonal atrx. n reste the egenvales have to be coted! Observatons: nverse teraton s chea, becase of trdagonal for Parallel and ndeendent nverse teraton for dfferent egenvales Fnd a good startng vector sch that we need only sall nber of teratons! 6

20 Mltle Relatvely Robst Reresentatons = MRRR Otlne of the algorth: Cote egenvale aroxaton λ wth hgh relatve accracy (e.g. Bsecton) Fnd the coln nber r of ( λ) - wth largest nor se bdagonal factorzatons = LDL. Perfor one ste of nverse teraton ( λ) z = e r MR 3 allows the cotaton of egenvectors wth hgh accracy (also for sall or close together egenvales) sng factorzatons: L + D + L + = LDL - σ. 7

21 Fnd otal coln k: Consder LDL v e wth v ( k) v k hen e LDL k k k e k k ek LDL k, k k k k here exst O(n) algorths for cotng all γ k factorzatons LDL - λ = L + D + L + = - D - -. based on For fndng the coln r of largest nor choose coln wth n γ k. 8

22 8.9 Seqental QR Algorth for cotng all Egenvales: Standard algorth for cotng egenars: QR-algorth Preste: ransfor A by Gvens or Hoseholder atrces to trdagonal for. a a a a 3 H G,3 * a3 a3 a33 * * * G, 3 a a 3 * * * * * * * * * * * * * * to elnate a 3 and a 3 Man dfference to QR-factorzaton: - se sbdagonal entry for elnatng eleents - Aly Q fro both sdes - Gves trdagonal atrx (or er Hessenberg for nonsyyetrc A). For better arallels se block Hoseholder lke n the QR-decooston. 9

23 QR-Algorth Frst ste: By Hoseholder atrces transfor A by eqvalence transforatons on trdagonal (er Hessenberg) for: A H*A*H = For the followng we asse A already trdagonal (er Hessenberg) Second ste: Cote QR-decooston of A, A = QR and relace A = A old by A new = RQ A new RQ ( Q A) Q Q AQ herefore A and A new have the sae egenvales Reeat these QR-stes ntl convergence aganst dagonal (er tranglar) atrx 3

24 8. woste rdagonalzaton Redce fll atrx to trdagonal (er Hessenberg) Seqental! For allowng better arallels redce atrx A to block-banded for, and then n a second ste to trdagonal for. Advantage: Frst ste allows block/blas3 oeratons and s good n arallel. second ste s chea; can be leented e.g. by MR 3. 3

25 Bothsded Hoseholder for rdagonalzaton Cote Hoseholder vector n order to elnate sbtrdagonal entres n the frst coln/row. Aly A (- H )A(- H ) = A ( H A) (A) H + 4 H ( H A) = = A ( H A+r H ) (A+r) H = = A y H -y H o redce BLAS oeratons work blockwse, A A Y H -Y H (BLAS3) bt stll frst A s needed (BLAS). 3

26 Block-Band redcton n the frst ste fnd QR decooston of sbblock A( + b : n, : n b )=A where b s the bandwdth and n b s a block sze. Cote QR decooston of black art A : Alyng (, Q H ) fro the left leads to tranglar for of black art. Alyng fro both sdes: Band strctre. Store Hoseholder vectors on ostons of new generated zeros. 33

27 D-Cyclc Data Dstrbton 4 x 4 Matrx on x rocessor array a a a 3 a 4 a a a 3 a 4 a 3 a 3 a 33 a 34 a 4 a 4 a 43 a 44 Advantage: better load balancng becase atrces and Hoseholder vectors are gettng saller. 34

28 Mltle Matrx Mltlcaton Cote H, k k for all k=,,,n wth n x n atrces,, N otal costs seqentally: N*n 3 here exst fast atrx-atrx algorths that are faster than n 3 (Strassen, gro-theoretc) Conjectre: O(n +ε ) 7

29 Block Coln Parallel... k Dstrbte 8 on k rocessors k together wth fll 7. k 7 8 (:, : n ) 7 8(:, n : n) 7 8(:, nk : n) Gves H 7,8 = 7 8 8

30 Block Coln Parallel... k H 78 Send fll 6 to all rocessors k. k 6 H 78 (:, : n ) 6 H78(:, n : n) 6 H78(:, nk : n) Gves H 6,8 =

31 Block Coln Parallel... k H 68 Send fll 5 to all rocessors k. k 5 H 68 (:, : n ) 5 H68(:, n : n) 5 H68(:, nk : n) Gves H 5,8 =

32 Block Coln Parallel... k H 8 Send fll to all rocessors k. k H 8 (:, : n ) H 8(:, n : n) H 8(:, nk : n) Gves H,8 = 6 7 8

33 Costs n Parallel:... k N- tes n * n/k = (N-)*n 3 / k For N atrces of n x n sze wth k rocessors. Esecally for 8 atrces and 4 rocessors: (7/4)*n 3 Concaton: n N

34 Parallel Prefx ree P P P 3 P 4 3

35 Parallel Prefx ree P P P 3 P 4 4

36 Parallel Prefx ree Costs: log(n)*n 3 wth N/ rocessors Esecally: 3*n 3 Seqentally: A lttle bt ore exensve than the colnwse ethod, bt less storage. 5